Spectral Criteria for Solvability of Boundary Value Problems and Positivity of Solutions of Time-Fractional Differential Equations

We investigate mild solutions of the fractional order nonhomogeneous Cauchy problem D(t)(alpha)u(t) = Au(t) +f (t) , t > 0 , where 0 < alpha < 1 . When ?? is the generator of a (T(t))(t >= 0) -semigroupon a Banach space X , we obtain an explicit representation of mild solutions of the above problem in terms of the semigroup. We then prove that this problem under the boundary condition u(0) =u(1)admits a unique mild solution for each f is an element of C([0,1]X] ;X if and only if the operator I-S-alpha( 1 ) is invertible. Here, we use the representation S-alpha(t) x-integral(0)8= phi(alpha)(S)T (St(alpha))xds t>0 in which phi(alpha) is a Wright type function. For the first order case, that is, alpha= 1 , the corresponding result was proved by Pruss in 1984. In case is a Banach lattice and the semigroup (T(t))(t >= 0) is positive, we obtain existence of solutions of the semilinear problem D(t)(alpha)u(t) = Au(t) +f (t) , t > 0 , where 0 < alpha < 1.